Document Type: Research Paper


Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran.


In this paper, we give three functors $\mathfrak{P}$, $[\cdot]_K$ and $\mathfrak{F}$ on the category of C$^\ast$-algebras. The functor $\mathfrak{P}$ assigns to each C$^\ast$-algebra $\mathcal{A}$ a pre-C$^\ast$-algebra $\mathfrak{P}(\mathcal{A})$ with completion $[\mathcal{A}]_K$. The functor $[\cdot]_K$ assigns to each C$^\ast$-algebra $\mathcal{A}$ the Cauchy extension $[\mathcal{A}]_K$ of $\mathcal{A}$ by a non-unital C$^\ast$-algebra $\mathfrak{F}(\mathcal{A})$. Some properties of these functors are also given.  In particular, we show that the functors $[\cdot]_K$ and $\mathfrak{F}$ are exact and the functor $\mathfrak{P}$ is normal exact.


Main Subjects

[1] W. Arveson, Notes on extensions of $C$*-algebras, Duke Math. J., 44 (1977), pp. 329-355.

[2] R.G. Bartle, The elements of real analysis. Second edition, John Wiley & Sons, New York-London-Sydney, 1976.

[3] B. Blackadar, Operator algebras. Theory of $C$*-algebras and Von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006.

[4] D.P. Blecher and C. Le Merdy, Operator algebras and their modules an operator space approach, London Mathematical Society Monographs. New Series, 30. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford, 2004.

[5] L.G. Brown, R.G. Douglas and P.A. Fillmore, Extensions of $C$*-algebras and $K$-homology, Ann. of Math., 105 (1977), pp. 265-324.

[6] R.C. Busby, Double centralizers and extensions of $C$*-algebras, Trans. Amer. Math. Soc., 132 (1968), pp. 79-99.

[7] J.B. Conway, A course in functional analysis, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1985.

[8] J. Dieudonne, Foundations of modern analysis. Enlarged and corrected printing, Pure Appl. Math., Vol. 10-I. Academic Press, New York-London, 1969.

[9] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), pp. 353-367.

[10] G.G. Kasparov, The operator $K$-functor and extensions of $C$*-algebras, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), pp. 571-636.

[11] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971.

[12] G.J. Murphy, $C$*-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.

[13] G.A. Reid, Epimorphisms and surjectivity, Invent. Math., 9 (1969/1970), pp. 295-307.

[14] J.J. Rotman, An introduction to homological algebra, Second edition. Universitext. Springer, New York, 2009.

[15] W. Rudin, Principles of mathematical analysis, Third edition. International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-DCsseldorf, 1976.

[16] K.W. Yang, Completion of normed linear spaces, Proc. Amer. Math. Soc., 19 (1968), pp. 801-806.